The Hecke-Kesten problem for some integrals (Q2857817)

The paper deals with the distribution of the function \(f(x)=\langle \alpha x\rangle\) where \(\langle x\rangle\) is the fractional part of the \(x.\) Let NEWLINE\[NEWLINE r(\alpha,n,I)=\#\{i: 0\leq i< n, \langle i\alpha\rangle \in I\}-n|I|NEWLINE\]NEWLINE The aim of the paper is to investigate (for fixed \(\alpha\)) bounded remainder sets \(I\) such that one has \(r(\alpha,n,I)\ll 1\). The Hecke-Kesten criterion for an interval to be a bounded remainder set is well known. Also there exists a general Oren's criterion for a set to be a bounded remainder set. But this criteria do not give any bound on \(r(\alpha,n,I)\). The author proves a new criterion for the set \(I=I_1\bigcup I_2\ldots\bigcup I_m,\) where \(I_k=[i_k;j_k)\) are intervals, to be a bounded remainder set. For such sets a new bound for \(r(\alpha,n,I)\) is obtained.